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<Title>Monoid</Title>
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<H1><A NAME="concept:Monoid"></A>
Monoid
</H1>

A <i>Monoid</i> is a concept that describes a simple kind of algebraic
system. A <b>monoid</b> consists of a set of elements <i>S</i>, a
binary operation, and an identity element. The C++ representation of a
monoid consists of a function object that implements the binary
operation, a set of objects that represent the elements of <i>S</i>,
and an object that represents the identity element.


<H3>Refinement of</H3>

The element type must be a model of <a
href="../../utility/Assignable.html">Assignable</a> and <a
href="../../utility/CopyConstructible.html">CopyConstructible</a>.
The function object type must be a model of <a
href="http://www.boost.org/sgi/stl/BinaryFunction.html">BinaryFunction</a>.

<h3>Notation</h3>

<Table>
<TR>
<TD><tt>X</tt></TD>
<TD>Is the element type of the Monoid.</TD>
</TR>

<TR>
<TD><tt>a, b</tt></TD>
<TD>are objects of type <tt>X</tt>.</TD>
</TR>

<TR>
<TD><tt>op</tt></TD>
<TD>Is the function object implementing the Monoid operation.</TD>
</TR>

<TR>
<TD><tt>i</tt></TD>
<TD>is an object of type <tt>X</tt> and is the identity element
for the Monoid.</TD>
</TR>


</table>

<h3>Valid Expressions</h3>

<Table border>

<tr>
<td><a name="sec:source"><TT>op(a,&nbsp;b)</TT></a></TD>
<TD>See below for semantics.<br>
Return type: <TT>X</TT>
</TD>
</TR>

<tr>
<TD><TT>a&nbsp;==&nbsp;b</TT></TD>
<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent
     the same element of <i>S</i>.<br>
Return type: <TT>bool</TT>
</TD>
</TR>

<tr>
<TD><TT>a&nbsp;!=&nbsp;b</TT></TD>
<TD>Returns true if <tt>a</tt> and <tt>b</tt> represent
     different elements of <i>S</i>.<br>
Return type: <TT>bool</TT>
</TD>
</TR>

</TABLE>

<P>

<H3>Invariants</H3>

<UL>
  <li>Closure<br>
     The result of <tt>op(a, b)</tt> is also an element of <i>S</i>.
  <li>Associativity<br>
     <tt>op(op(a, b), c) == op(a, op(b, c))</tt>
  <li>Definition of Identity Element<br>
       <tt>op(a, i) == a</tt>
</UL>

<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>,
Indiana University (<A
HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br>
<A HREF="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</A>, Indiana University (<A HREF="mailto:llee@cs.indiana.edu">llee@cs.indiana.edu</A>)<br>
<A HREF="https://homes.cs.washington.edu/~al75">Andrew Lumsdaine</A>,
Indiana University (<A
HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>)
</TD></TR></TABLE>

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